\section{State Estimation}
\label{sec:mpc:estimation}

All controllers are assumed to have full state information available to generate their prediction matrices.
The state estimate is obtained using an observer with a static gain $M$ as follows:

\begin{equation}
  \begin{split}
    \vc{\hat{x}}_{k|k-1}^a & = A_{k-1}^a \left(\Delta \vc{\hat{x}}_{k-1}^a - \vc{u}_{\text{r},k-2}\right) + B_{k-1}^a \Delta \vc{u}_{k-1}\\
    \vc{\hat{x}}_{k} & = \vc{\hat{x}}_{k|k-1}^a + M \left( \vc{y}_k-\vc{y}_{k-1} - C_k^a \vc{\hat{x}}_{k|k-1}^a \right)
  \end{split}
\end{equation}

\noindent where $\vc{\hat{x}}_{k|k-1}^a$ is the a priori estimate of $\vc{\hat{x}}_k^a$ and $\vc{u}_{\text{r},k-2}$ is the recycle valve input about which the system was linearized, as defined in \eqref{eq:mpc:augmented-state-eqs}.

The observer gain $M$ was computed offline using a Kalman filter based on the linearized model at the design point of the system.
It is assumed to be constant and not updated at each time step with the new linearized model for simplicity.
Additionally, in the physical installation, the states are measured directly using sensors and the observer gain is not crucial.

